How to Identify and Use Consecutive Interior Angles in Geometry
Also known as co interior angle, consecutive angles are vertically opposite angles that are equal to each other. That being said, when two lines are cut by a transversal, the pair of angles on one side of the transversal and on the interior of the two lines are known as the consecutive interior angles. In the figure shown below, angles 3 and 5 are said to be consecutive interior angles.
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Alternate Interior Angles
Angles on the opposite side of the transversal are called alternate interior angles.
Example: angle 1 and angle 2 are alternate interior angles.
Using the concept of angles and their results we can prove that angles in a triangle add to 180°
Draw two parallel lines E and F
Since alternate interior angles are equal and pairs of angle 1 and angle 2 form alternate interior angles, therefore, angle 1 and angle 2 are equal. The same is for angle 3 and angle 4.
Since angle 1, angle 3 and angle 5 are around the straight line, therefore they add up to 180°
From this, we can conclude that angles in a triangle add to 180°
Since angle 1 = angle 2 and angle 3 = angle 4, therefore angle 5 + angle 4 + angle 2 = 180°
Also, angle 6 is the exterior angle of the triangle and the exterior angle of the triangle is equal to the sum of the two interior angles.
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Properties of Consecutive Interior Angles
You know the co interior angles definition. Now refer to the following key properties of consecutive interior angles:
Consecutive interior angles are formed when two lines are cut by the transversal the pair of angles formed inside the lines on one side.
Consecutive interior angles are called supplementary when the two lines are parallel, and the pair of angles add to 180°
The angles are called “consecutive” because they follow each other consecutively on the same side of the line.
The pair of angles which lie on the exterior of the transversal on the same side are called consecutive exterior angles.
Angles formed inside the parallel lines are cut by the transversal on the same side are called co-interior angles. Co-interior angles are supplementary but they are not equal when the lines are parallel.
Consecutive Interior Angles Theorem- Proof
See the figure shown below.
Since we are already aware that the two lines are parallel, therefore we have:
∠1=∠5 (corresponding angles)
From the aforementioned two equations, we get
∠1+∠4=180⁰
In the same manner, we can show that
∠2+∠3=180⁰
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Converse of Consecutive Interior Angle Theorem
Proof
Suppose that
∠ 1+∠4=180⁰→(1)
Because ∠5 and ∠4 forms a linear pair,
∠5+∠4 =180∘ →(2)
From the equations (1) and (2),
∠1=∠5
Therefore, a pair of corresponding angles are equivalent that can only take place if the two lines are parallel.
Thus, the converse of consecutive interior angle theorems is proven.
Solved Examples
Example:
Are the following lines naming l and m in the figure below parallel?
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Solution:
In the figure, 125o and 60o are the consecutive interior angles given that they are supplementary.
But
125 + 60 =185o
Hence, 125o and 60o are NOT supplementary angles.
Therefore, the given lines are NOT parallel as per the "Consecutive Interior Angle Theorem,”
Hence, l and m are NOT parallel.
Example:
Consider the given figure, in which L1 and L2 are the parallel lines. Calculate the value of ∠C?
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Solution:
Through C, draw a line parallel to L1 and L2, as given in the figure.
We have:
∠x = ∠β = 60 degrees (alternate interior angles)
∠y = 180o – 120 degrees (consecutive interior angles)
∠y = 60degrees
Therefore, we obtain
∠C = ∠x + ∠y = 120degrees
Hence, ∠C=120 degrees
Fun Facts
The eight angles together form four pairs of corresponding angles.
Corresponding angles are in congruence to one another if the two lines are parallel.
Vertical angles are formed by only two intersecting lines and they are non-adjunct angles.
FAQs on Consecutive Interior Angles Explained
1. What are consecutive interior angles?
Consecutive interior angles are a pair of angles that are located on the same side of the transversal and are in between the two lines it intersects. They are also commonly known as co-interior angles or same-side interior angles. For two angles to be considered consecutive interior angles, they must:
- Lie in the interior region (between the two lines).
- Be on the same side of the transversal line.
- Have different vertices.
2. What is the Consecutive Interior Angles Theorem?
The Consecutive Interior Angles Theorem states that if two parallel lines are intersected by a transversal, then the pairs of consecutive interior angles formed are supplementary. This means that their sum is always equal to 180 degrees. This theorem is a fundamental property used in geometry to solve problems involving parallel lines.
3. How can you find the measure of a consecutive interior angle?
To find the measure of a consecutive interior angle, you must first know if the two lines are parallel. If the lines are parallel and you know the measure of one angle in a pair (let's say it is 'x'), you can find the other by subtracting it from 180°. The formula is: Angle 2 = 180° - Angle 1. For example, if one consecutive interior angle is 110°, its pair must be 180° - 110° = 70°.
4. Are consecutive interior angles always supplementary?
No, they are not. This is a common misconception. Consecutive interior angles are only supplementary if the two lines intersected by the transversal are parallel. If the lines are not parallel, the angles are still identified as consecutive interior angles by their position, but their sum will not be 180°.
5. What is the difference between consecutive interior angles and alternate interior angles?
The main difference lies in their position and their properties when lines are parallel.
- Position: Consecutive interior angles are on the same side of the transversal, while alternate interior angles are on opposite (alternating) sides of the transversal.
- Relationship (with parallel lines): Consecutive interior angles are supplementary (add up to 180°). In contrast, alternate interior angles are congruent (equal in measure).
6. What is the importance of the Converse of the Consecutive Interior Angles Theorem?
The converse of the theorem is crucial for proving that two lines are parallel. It states: If a transversal intersects two lines such that a pair of consecutive interior angles is supplementary (their sum is 180°), then the two lines are parallel. While the original theorem helps find angle measures when lines are known to be parallel, the converse helps establish that the lines are parallel in the first place.
7. Can you provide a real-world example of consecutive interior angles?
Yes. A perfect real-world example is a rectangular window frame or a door frame. The two vertical sides of the frame are parallel lines, and the horizontal top or bottom piece acts as a transversal. The two angles formed on the inside at each top corner are consecutive interior angles. Since the sides are parallel, these angles are supplementary (90° + 90° = 180°), ensuring the frame is perfectly rectangular.
